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First of all, you must be prepared to such trip - it is 4 days long from Moscow to Mongolia (with changes), and you should get some stuff like electronic books or films. Also you should think about possibilities of recharging your electronical devices - charges are available not in all trains and not at each stations.

After those choices, I recommend you to decide what class you will travel:

High class ('Spalniy' in Russian)

1st class ('Kupe' in Russian)

2d class ('Platzkart' in Russian)

High class is very expensive (about $700), with high quality service. In railroad car you travel in rooms for only two persons. It is very comfortable, calm and quiet place.

1st class is much cheaper (about $250), rooms are for 4 persons, and many services are not free. This variant is also comfortable, and here you can make a good trip.

2d class is cheapest ($150) and crowded. Rooms in railroad car are not closing, and on other side additional places are situated. This type of travel is noisy, but if you are searching some kind of communication, you should go here.

From weather point of view, a definitely not recommend you to travel at the middle of the summer (from June, 20 to the July, 20) - this is very hot time of the year, and not every railroad car provides possibilities to get fresh air (even in High class).
Also you should not travel at the middle of the winter (in December and January) - this is very cold time of the year, days are short, and you will not get any joy from travel.

Also, you should check the number of room when you get a ticket - some of them (this is depend on train types and class you've selected) have no way to open a window and refresh to room (because there are emergency exits there).

Try to get cheapest tickets by season prices (sorry, only in Russian - generaly it is March, April and October months) and avoid public holidays (May, 1st and May, 9th, for example)

Thanks for this excellent, comprehensive answer! I will likely post a few more questions on this topic (but will break them into separate posts to make them more accessible to other users)
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The Travelling ChickenJul 11 '11 at 15:56